Application of approximation theory by nonlinear manifolds in Sturm-Liouville inverse problems

Abstract

We give here some negative results in Sturm-Liouville inverse theory, meaning that we cannot approach any of the potentials with $m+1$ integrable derivatives on ${\mathbb{R}}^{+}$ by an $\omega$-parametric analytic family better than order of $(\omega \ln \omega {)}^{-(m+1)}$. Next, we prove an estimation of the eigenvalues and characteristic values of a Sturm-Liouville operator and some properties of the solution of a certain integral equation. This allows us to deduce from [Henkin-Novikova] some positive results about the best reconstruction formula by giving an almost optimal formula of order of ${\omega }^{-m}$.

arXiv comment40 pages

References in corpus (1)

• Application of approximation theory by nonlinear manifolds in Sturm-Liouville inverse problems

Cited by in corpus (2)

• Application of approximation theory by nonlinear manifolds in Sturm-Liouville inverse problems
• An approximation formula for holomorphic functions by interpolation on the ball